
Remark 10.68.7 (Koszul regular sequences). In the paper [Kabele] the author introduces two more regularity conditions for sequences $x_1, \ldots , x_ r$ of elements of a ring $R$. Namely, we say the sequence is Koszul-regular if $H_ i(K_{\bullet }(R, x_{\bullet })) = 0$ for $i \geq 1$ where $K_{\bullet }(R, x_{\bullet })$ is the Koszul complex. The sequence is called $H_1$-regular if $H_1(K_{\bullet }(R, x_{\bullet })) = 0$. If $R$ is a local ring (possibly non-Noetherian) and the sequence consists of elements of the maximal ideal, then one has the implications regular $\Rightarrow$ Koszul-regular $\Rightarrow$ $H_1$-regular $\Rightarrow$ quasi-regular. By examples the author shows that these implications cannot be reversed in general. We introduce these notions in more detail in More on Algebra, Section 15.29.

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